\Header{findInterval}{Find Interval Numbers or Indices}
\keyword{arith}{findInterval}
\keyword{utilities}{findInterval}
\begin{Description}\relax
Find the indices of \code{x} in \code{vec}, where \code{vec} must be
sorted (non-decreasingly); i.e., if \code{i <- findInterval(x,v)},
we have \eqn{v_{i_j} \le x_j < v_{i_j + 1}}{v[i[j]] <= x[j] < v[i[j] + 1]}
where \eqn{v_0 := -\infty}{v[0] := - Inf},
\eqn{v_{N+1} := +\infty}{v[N+1] := + Inf}, and \code{N <- length(vec)}.
At the two boundaries, the returned index may differ by 1, depending
on the optional arguments \code{rightmost.closed} and \code{all.inside}.\end{Description}
\begin{Usage}
\begin{verbatim}
findInterval(x, vec, rightmost.closed = FALSE, all.inside = FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] numeric.
\item[\code{vec}] numeric, sorted (weakly) increasingly, of length \code{N},
say.
\item[\code{rightmost.closed}] logical; if true, the rightmost interval,
\code{vec[N-1] .. vec[N]} is treated as \emph{closed}, see below.
\item[\code{all.inside}] logical; if true, the returned indices are coerced
into \eqn{\{1,\dots,N-1\}}{}, i.e., 0 is mapped to 1 and \eqn{N}{} to
\eqn{N-1}{}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The function \code{findInterval} finds the index of one vector \code{x} in
another, \code{vec}, where the latter must be non-decreasing.  Where
this is trivial, equivalent to \code{apply( outer(x, vec, ">="), 1, sum)},
as a matter of fact, the internal algorithm uses interval search
ensuring \eqn{O(n \log N)}{O(n * log(N))} complexity where
\code{n <- length(x)} (and \code{N <- length(vec)}).  For (almost)
sorted \code{x}, it will be even faster, basically \eqn{O(n)}{}.

This is the same computation as for the empirical distribution
function, and indeed, \code{findInterval(t, sort(X))} is
\emph{identical} to \eqn{n F_n(t; X_1,\dots,X_n)}{n * Fn(t;
X[1],..,X[n])} where \eqn{F_n}{Fn} is the empirical distribution
function of \eqn{X_1,\dots,X_n}{X[1],..,X[n]}.

When \code{rightmost.closed = TRUE}, the result
for \code{x[j] = vec[N]} (\eqn{ = \max(vec)}{}), is \code{N - 1} as for
all other values in the last interval.\end{Details}
\begin{Value}
vector of length \code{length(x)} with values in \code{0:N} where
\code{N <- length(vec)}, or values coerced to \code{1:(N-1)} iff
\code{all.inside = TRUE} (equivalently coercing all x values
\emph{inside} the intervals).\end{Value}
\begin{Author}\relax
Martin Maechler\end{Author}
\begin{SeeAlso}\relax
\code{\Link{approx}(*, method = "constant")} which is a
generalization of \code{findInterval()}, \code{\Link{ecdf}} for
computing the empirical distribution function which is (up to a factor
of \eqn{n}{}) also basically the same as findInterval(.).\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
N <- 100
X <- sort(round(rt(N, df=2), 2))
tt <- c(-100, seq(-2,2, len=201), +100)
it <- findInterval(tt, X)
tt[it < 1 | it >= N] # only first and last are outside range(X)
\end{ExampleCode}
\end{Examples}

